Why 2 + 2 = 4

Date: 10/27/95 at 21:21:47
From: Pawel Gora
Subject: Question

This is an original question I have been asked by a student:

WHY is 2+2=4 ?

Best Regards
   Pawel Gora

Date: 10/28/95 at 12:11:3
From: Doctor Ken
Subject: Re: Question

Hello there!

This question is actually pretty difficult, and involves some very 
fundamental, deep mathematics.  I think I read somewhere that a couple 
of people named Whitehead and Russel wrote a book (a few volumes, I 
believe) that started with a few basic axioms about set theory, and 
started deriving the mathematics that we know today in a VERY rigorous way.  
I think it took them hundreds of pages before they were able to show that 
1+1=2.  So there's a little more than I can really tell you here (and to 
tell the truth, more than I know myself).

On the other hand, I can tell you a few things about the problem.  The 
really hard thing is to show that 1+1=2.  Once you have that, it's 
relatively easy to show that 2+2=4.  What it means for 1+1 to equal 2 is 
that they're the SAME THING.  Think about this for a moment.  That means 
that if we do the same thing to both of them, we should get the same 
result.  So let's do that.  Let's take both sides of the equation and 
add them to themselves.  We can think about that as just adding the two 
equations, like this:

      1+1 = 2
    + 1+1 = 2
    ---------
  1+1+1+1 = 2+2

Now think about the left side of the equation.  What is "4?"  Well, it's 
just a name for the "fourth" (i.e., this many: * * * *) item in the 
following list: 

1
1+1
1+1+1
1+1+1+1
1+1+1+1+1
...

So "4" just means "start counting by ones until you reach the 'fourth' 
item."


So we have this:

      1+1 = 2
    + 1+1 = 2
    ---------
  1+1+1+1 = 2+2
        4 = 2+2

QED.

-Doctor Ken,  The Geometry Forum

Archivists' Note: Mathematically sophisticated folks may also want to see 

   Why does 2+2 = 4?
   http://mathforum.org/library/drmath/view/51521.html   .